The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

Song, Tianming; Main, Alex; Scovazzi, Guglielmo; Ricchiuto, Mario

doi:10.1016/j.jcp.2018.04.052

Cited by 66 publications

(45 citation statements)

References 48 publications

(71 reference statements)

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“…In the following, we recall the full-order Shifted Boundary Method (SBM) for the steady Stokes equations following [32]. The Shifted Boundary Method is an embedded (immersed, non-conformal) finite element method, which relies on a Nitsche-type approach for the weak imposition of Dirichlet boundary conditions [32,33,48]. Weak boundary conditions require less complicated data structures with respect to strongly imposed boundary conditions, and, for this reason, the SBM relies on the Nitsche approach.…”

Section: 2mentioning

confidence: 99%

A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow

Karatzas

Stabile

Nouveau

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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Section: 2mentioning

confidence: 99%

A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow

Karatzas

Stabile

Nouveau

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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“…Weak SBM formulation. In this subsection we briefly recall the SBM formulation which was originally presented in [17,18,25]. In what follows, we denote byΓ the surrogate boundary composed of the edges/faces of the mesh that are the closest to the true boundary Γ .…”

Section: 2mentioning

confidence: 99%

A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries

Karatzas

Stabile

Atallah

et al. 2019

IUTAM Symposium on Model Order Reduction of Coupled Systems, Stuttgart, Germany, May 22–25, 2018

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A model order reduction technique is combined with an embedded boundary finite element method with a POD-Galerkin strategy. The proposed methodology is applied to parametrized heat transfer problems and we rely on a sufficiently refined shape-regular background mesh to account for parametrized geometries. In particular, the employed embedded boundary element method is the Shifted Boundary Method (SBM), recently proposed in [17]. This approach is based on the idea of shifting the location of true boundary conditions to a surrogate boundary, with the goal of avoiding cut cells near the boundary of the computational domain. This combination of methodologies has multiple advantages. In the first place, since the Shifted Boundary Method always relies on the same background mesh, there is no need to update the discretized parametric domain. Secondly, we avoid the treatment of cut cell elements, which usually need particular attention. Thirdly, since the whole background mesh is considered in the reduced basis construction, the SBM allows for a smooth transition of the reduced modes across the immersed domain boundary. The performances of the method are verified in two dimensional heat transfer numerical examples.

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“…The proposed work stems from the Shifted Boundary Method (SBM), an embedded method that has been recently developed and applied in the context of the Laplace and Stokes problems [28], the Navier-Stokes equations [29] and hyperbolic systems [46]. The SBM leverages a surrogate/approximate interface representation, by which the location where boundary conditions are applied is shifted from the true to the surrogate interface, and, most importantly, a Taylor expansion is used to modify the value of boundary conditions, with the goal of preventing a reduction in the convergence rates of the overall formulation.…”

Section: Introductionmentioning

confidence: 99%

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

Nouveau

Ricchiuto

Scovazzi

2019

Journal of Computational Physics

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We propose an extension of the embedded boundary method known as "shifted boundary method" to elliptic diffusion equations in mixed form (e.g., Darcy flow, heat diffusion problems with rough coefficients, etc.). Our aim is to obtain an improved formulation that, for linear finite elements, is at least second-order accurate for both flux and primary variable, when either Dirichlet or Neumann boundary conditions are applied. Following previous work of Nishikawa and Mazaheri in the context of residual distribution methods, we consider the mixed form of the diffusion equation (i.e., with Darcy-type operators), and introduce an enrichment of the primary variable. This enrichment is obtained exploiting the relation between the primary variable and the flux variable, which is explicitly available at nodes in the mixed formulation. The proposed enrichment mimics a formally quadratic pressure approximation, although only nodal unknowns are stored, similar to a linear finite element approximation. We consider both continuous and discontinuous finite element approximations and present two approaches: a non-symmetric enrichment, which, as in the original references, only improves the consistency of the overall method; and a symmetric enrichment, which enables a full error analysis in the classical finite element context. Combined with the shifted boundary method, these two approaches are extended to high-order embedded computations, and enable the approximation of both primary and flux (gradient) variables with second-order accuracy, independently on the type of boundary conditions applied. We also show that the the primary variable is third-order accurate, when pure Dirichlet boundary conditions are embedded. similar indicator function. The main idea behind the IBM is to solve the model equations on the entire domain, and to impose the boundary conditionss via a forcing term. The method was originally designed on Cartesian grids and its development has been an active field of research for some decades now. Two exhaustive reviews were written by Mittal and Iaccarino in 2005 [34], and Sotiropoulos and Yang in 2014 [47]. The main drawback of the IBM is the accuracy of boundary conditions, which are most often only first-order accurate. Some strategies have been proposed in the past to increase the IBM accuracy, at the expense of more involved implementations [18,17,26,27], or to compensate low-order accuracy by means of mesh adaptation [38,6,19].A similar objective is pursued by Embedded Boundary Methods, which still use an immersed description of bodies and boundaries, but solve the model equations only in the regions of interest. Within a finite element context, the cut-cell and cutFEM approach [39,15,20,10,24,48] are most commonly utilized. These techniques usually combine a weak enforcement of the boundary conditions with a XFEM strategy. For example, in CutFEMs, the boundary is reconstructed as the intersection between the boundary surface and the embedding (underlaying) grid. Approaches of this kind often require complex...

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The shifted boundary method for hyperbolic systems: Embedded domain computations of linear waves and shallow water flows

Cited by 66 publications

References 48 publications

A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow

A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow

A Reduced Order Approach for the Embedded Shifted Boundary FEM and a Heat Exchange System on Parametrized Geometries

High-order gradients with the shifted boundary method: An embedded enriched mixed formulation for elliptic PDEs

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